Jack had a bag of $128$ apples. He sold $25\% $ of them to Jill. Next he sold $25\% $ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then? $\text{(A)}\ 7 \qquad \text{(B)}\ 63 \qquad \text{(C)}\ 65 \qquad \text{(D)}\ 71 \qquad \text{(E)}\ 111$

On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.

The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle. (A. Zaslavsky)

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that 1) neither digit is $0$, and 2) the units digit is a multiple of the tens digit? [b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer $k$ such that Mirabel could have had $k$ leftover candies. [b]p3[/b] Callie picks two distinct numbers from $\{1, 2, 3, 4, 5\}$ at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is $p$. $p$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$. Find $a + b$.

There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube? $ \textbf{(A)}\ 54 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 96 $

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

Let $ ABC$ be an acute-angled triangle; $ AD$ be the bisector of $ \angle BAC$ with $ D$ on $ BC$; and $ BE$ be the altitude from $ B$ on $ AC$. Show that $ \angle CED > 45^\circ .$ [b][weightage 17/100][/b]

Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$

Initially, there are $3^{80}$ particles at the origin $(0,0)$. At each step the particles are moved to points above the $x$-axis as follows: if there are $n$ particles at any point $(x,y)$, then $\Bigl \lfloor \frac{n}{3} \Bigr\rfloor$ of them are moves to $(x+1,y+1)$, $\Bigl \lfloor \frac{n}{3} \Bigr\rfloor$ are moved to $(x,y+1)$ and the remaining to $(x-1,y+1)$, For example, after the first step, there are $3^{79}$ particles each at $(1,1),(0,1)$ and $(-1,1)$. After the second step, there are $3^{78}$ particles each at $(-2,2)$ and $(2,2)$, $2 \times 3^{78}$ particles each at $(-1,2)$ and $(1,2)$, and $3^{79}$ particles at $(0,2)$. After $80$ steps, the number of particles at $(79,80)$ is:

[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("$A$",(0,0),SW); label("$B$",(1,0),SE); label("$C$",(1,1),NE); label("$D$",(0,1),NW); label("$E$",(.5,sqrt(3)/2),E); label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram[/asy] Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is $\textbf{(A) }1\qquad\textbf{(B) }\frac{\sqrt{2}}{2}\qquad\textbf{(C) }\frac{\sqrt{3}}{2}$ $\qquad\textbf{(D) }4-2\sqrt{3}\qquad \textbf{(E) }\frac{1}{2}+\frac{\sqrt{3}}{4}$

If $a,b,c\in\mathbb N\setminus\{0,1,2,3\}$ then prove: $$b^2\cdot\sqrt[a]a+c^2\cdot\sqrt[b]b+a^2\cdot\sqrt[c]c\ge48\sqrt2$$ [i]Proposed by Daniel Sitaru[/i]

Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$. Prove that for all $A \ge 1$, there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$, \[|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.\]

Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$

Compute the number of zeros at the end of $2015!$.

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.

Let $A$ and $B$ be two cubes. Numbers $1,2,...,14$, are assigned in any order, to the faces and vertices of cube $A$. Then each edge of cube $A$ is assigned the average of the numbers assigned to the two faces that contain it. Finally assigned to each face of the cube $B$ the sum of the numbers associated with the vertices, the face and the edges on the corresponding face of cube $A$. If $S$ is the sum of the numbers assigned to the faces of $B$, find the largest and smallest value that $S$ can take.

Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Karagumruk-Adalet is Karagumruk. In how many ways one can order these five teams such that each team except the last, defeated the next team? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $a_1, a_2,\dots$ be an infinite sequence of positive real numbers such that for each positive integer $n$ we have \[\frac{a_1+a_2+\cdots+a_n}n\geq\sqrt{\frac{a_1^2+a_2^2+\cdots+a_{n+1}^2}{n+1}}.\] Prove that the sequence $a_1,a_2,\dots$ is constant. [i]Proposed by Alex Zhai[/i]

Given that \begin{align*} x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523, \text{ and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4} \text{ when } n \geq 5, \end{align*} find the value of $x_{531}+x_{753}+x_{975}$.

Let $S$ be a set with a binary operation $\ast$ such that 1) $a \ast(a\ast b)=b$ for all $a,b\in S$. 2) $(a\ast b)\ast b=a$ for all $a,b\in S$. Show that $\ast$ is commutative and give an example where $\ast$ is not associative.

Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.

Consider a cuboid$ ABCDA'B'C'D'$ (where $ABCD$ is a rectangle and $AA' \parallel BB' \parallel CC' \parallel DD'$) with $AA' = d$, $\angle ABD' = \alpha, \angle A'D'B = \beta$. Express the lengths x = $AB$, $y = BC$ in terms of $d$ and (acute) angles $\alpha, \beta$. Discuss condition of solvability.

The following operation is allowed on the positive integers: if a number is even, we can divide it by $2$, otherwise we can multiply it by a power of $3$ (different from $3^0$) and add $1$. Prove that we can reach $1$ from any starting positive integer $n$.

Let us call a convex hexagon $ABCDEF$ [i]boring [/i] if $\angle A+ \angle C + \angle E = \angle B + \angle D + \angle F$. a) Is every cyclic hexagon boring? b) Is every boring hexagon cyclic?